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Decimal expansion of the radius of convergence of the g.f. of A106336; equals constant A106333 divided by constant A106334.
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%I #13 Oct 15 2023 19:07:11

%S 3,2,2,6,2,7,6,3,2,6,9,2,1,9,1,1,3,3,0,9,6,9,8,7,1,3,8,6,7,3,9,8,3,0,

%T 2,3,3,2,2,9,0,4,2,4,3,7,4,6,7,1,7,4,5,2,1,6,0,5,6,2,0,9,1,2,4,5,5,4,

%U 8,6,2,6,7,4,1,1,1,5,0,6,4,9,7,4,7,1,2,3,7,3,9,9,1,2,2,1,4,7,8,5,3,7,1,9,0

%N Decimal expansion of the radius of convergence of the g.f. of A106336; equals constant A106333 divided by constant A106334.

%C The g.f. of A106336 equals (1/x)*Series_Reversion( x*eta(x)/eta(x^2)^2 ).

%C This constant is very close to 2^(3/2) / (3*sqrt(e*Pi)) = 0.3226276326921911330637735905807475397715626276499133673167401123748... - _Vaclav Kotesovec_, Aug 02 2017

%F Constant equals the ratio x/F(x) evaluated at the constant x that satisfies: F(x) - x*F'(x) = 0, where F(x) = Sum_{n>=0} x^(n*(n+1)/2).

%e x/F(x)=0.322627632692191133096987138673983023322904243746717452160562...

%e where F(x) = 1 + x + x^3 + x^6 + x^10 + x^15 + x^21 + x^28 + ...

%e so F(x) = 1.9873697211846841452692897833444126... (A106334)

%e at x = 0.6411803884299545796456448886283011... (A106333).

%t digits = 105; x0 = x /. FindRoot[ Sum[(1 - n*(n+1)/2)*x^(n*(n+1)/2), {n, 0, digits}], {x, 1/2}, WorkingPrecision -> digits+5]; f[x_] := EllipticTheta[2, 0, Sqrt[x]]/(2*x^(1/8)); x0/f[x0] // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Mar 05 2013 *)

%o (PARI) A106333=solve(x=.6,.7,sum(n=0,100,(1-n*(n+1)/2)*x^(n*(n+1)/2))); A106334=sum(n=0,100, A106333^(n*(n+1)/2)); A106335=A106333/A106334

%Y Cf. A106333, A106334, A106336.

%K cons,nonn

%O 0,1

%A _Paul D. Hanna_, Apr 29 2005